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I'm trying to understand the phase-number uncertainty relation for superconductors, \begin{align} \Delta N \Delta \varphi \gtrsim 1. \end{align}

In particular, I'm trying to understand if it holds for the states \begin{align} \vert \psi_\varphi \rangle &= \prod_k (u_k + e^{i\varphi} \nu_k \hat{c}^{\dagger}_{k \uparrow}\hat{c}^{\dagger}_{-k \downarrow}) \vert \psi_0 \rangle \\ \vert \psi_N \rangle &= \int_0^{2\pi} e^{iN\varphi/2} \vert \psi_\varphi \rangle \end{align} of well defined phase and well defined particle number.

It seems that, for the first case $\Delta N$ is finite (estimated by Tinkham to be $\approx 10^9$ - for a macroscopic classic superconductor, I believe), while $\Delta \varphi$ is zero (since $\varphi$ is well defined). For the second case, on the other hand, I would say $\Delta N = 0$ and $\Delta_\varphi = 2 \pi$. In both cases, the product of uncertainties is zero. But if that is the case, the uncertainty relation above would be simply wrong.

Is there a flaw in my train of thought? If not, what is the usefulness of the uncertainty relation above if it does not apply to the two most basic states in this context?

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    $\begingroup$ This 10^9 looks obscure. Where would any scale come in? Also, since $\Delta\varphi<\2pi$, this statement only makes sense as long as $\Delta\varphi\ll 2\pi$. $\endgroup$ Jul 23 at 22:53
  • $\begingroup$ The $10^9$ comes from $\Delta N \approx (\langle N \rangle T_c/T_F)^{1/2}$, where $\langle N \rangle \approx 10^{24}$. The scale comes from physical parameters of usual superconductors and Avogadro's number, I suppose. About your second point, which statement are you referring to? $\endgroup$ Jul 23 at 23:55
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    $\begingroup$ Ok, so the 10^9 has nothing to do with the formula above, but with a specific physical situation it refers to. Regarding point 2, the uncertainty relation. $\endgroup$ Jul 24 at 0:52
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    $\begingroup$ This is related to the problem of properly defining a self-adjoint phase operator. See for instance physics.stackexchange.com/q/338044/36194 $\endgroup$ Jul 24 at 13:29
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    $\begingroup$ Additional note: there are other discussions on phase operator throughout the site although I don't recall that they address the exact problem of $\Delta N=0$ that you are referring to. $\endgroup$ Jul 24 at 14:45

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